1
00:00:00,310 --> 00:00:01,998
Let's recycle sequences.

2
00:00:01,998 --> 00:00:02,927
[SOUND].

3
00:00:02,927 --> 00:00:08,139
[MUSIC].

4
00:00:08,139 --> 00:00:10,340
If I've got a sequence in hand, I can

5
00:00:10,340 --> 00:00:13,450
build a new sequence out of that old
sequence.

6
00:00:13,450 --> 00:00:15,080
Well, here's a sequence to start with.

7
00:00:15,080 --> 00:00:18,900
It's a sequence, the first three terms of
which, are all 1.

8
00:00:18,900 --> 00:00:22,888
So a sub 0, a sub 1, a sub 2 are all equal
to 1.

9
00:00:22,888 --> 00:00:28,540
And then to compute each subsequent term I
just add up the three previous terms.

10
00:00:28,540 --> 00:00:29,760
Let's see some terms.

11
00:00:29,760 --> 00:00:32,980
Well, here are some terms in this
sequence.

12
00:00:32,980 --> 00:00:38,910
The first three terms are all 1.
1, 1, 1, a sub zero, a sub 1, a sub 2.

13
00:00:38,910 --> 00:00:43,080
Each subsequent term is computed using
this recursive formula.

14
00:00:43,080 --> 00:00:46,500
What this formula tells me to do is to add
up the previous three terms.

15
00:00:46,500 --> 00:00:52,570
So this 3 is coming from 1 plus 1 plus 1.
This 5 is coming from 3 plus 1 plus 1.

16
00:00:52,570 --> 00:00:58,054
This 9 is coming from 5 plus 3 plus 1.
This 17 is coming from 9 plus 5 plus

17
00:00:58,054 --> 00:00:59,840
3, and so on.

18
00:00:59,840 --> 00:01:03,350
Now this might seem a little bit like the
Fibonacci sequence, except instead

19
00:01:03,350 --> 00:01:07,390
of adding up the previous 2 terms, I'm
adding up the previous 3 terms.

20
00:01:07,390 --> 00:01:08,765
So, to be a little bit funny,

21
00:01:08,765 --> 00:01:11,720
people sometimes call this the Tribonacci
sequence.

22
00:01:11,720 --> 00:01:15,940
I can build a new sequence out of the
Tribonacci sequence.

23
00:01:15,940 --> 00:01:19,862
So starting with this sequence, I could
build a new sequence,

24
00:01:19,862 --> 00:01:23,340
sequence I'll call b, by referring to the
terms in this

25
00:01:23,340 --> 00:01:28,040
Tribonacci sequence.
Let's see some terms of this sequence b.

26
00:01:28,040 --> 00:01:30,440
Here is the sequence a sub n.

27
00:01:30,440 --> 00:01:32,950
All right.
It goes 1, 1, 1, 3, 5.

28
00:01:32,950 --> 00:01:35,020
This is the Tribonacci sequence.

29
00:01:35,020 --> 00:01:38,700
And here is the beginning of the sequence
b sub n.

30
00:01:38,700 --> 00:01:40,876
According to this rule, b sub n is just

31
00:01:40,876 --> 00:01:45,050
the ratio of neighboring terms in the
tribonacci sequence.

32
00:01:45,050 --> 00:01:47,930
So, for example, this term here,

33
00:01:47,930 --> 00:01:53,402
which is b sub 0, b sub 1, b sub 2, b sub
3, is the ratio of a sub 4

34
00:01:53,402 --> 00:01:58,986
and a sub 3.
Or this term here, which is b sub 4, is

35
00:01:58,986 --> 00:02:05,060
the ratio of a sub 5 and a sub 4.
The fractions are sort of confusing.

36
00:02:05,060 --> 00:02:09,210
So we can replace these fractions with
decimal approximations.

37
00:02:09,210 --> 00:02:11,010
Instead of looking at these fractions.

38
00:02:11,010 --> 00:02:13,140
Here are some of the terms of b sub n,

39
00:02:13,140 --> 00:02:17,600
but written out in terms of their decimal
approximations.

40
00:02:17,600 --> 00:02:19,470
Let's look at these approximations.

41
00:02:19,470 --> 00:02:22,996
Remember, these are decimal approximations
to

42
00:02:22,996 --> 00:02:26,605
neighboring terms in the Tribonacci
sequence.

43
00:02:26,605 --> 00:02:30,610
And as we go out further and further in
the Tribonacci sequence.

44
00:02:30,610 --> 00:02:31,559
Do you notice something?

45
00:02:32,660 --> 00:02:35,358
Well, in light of this numeric evidence,
it

46
00:02:35,358 --> 00:02:38,127
certainly looks like this sequence, b sub
n,

47
00:02:38,127 --> 00:02:42,600
has a limit.
And this limit is about 1.839.

48
00:02:42,600 --> 00:02:45,400
So that raises a question.

49
00:02:45,400 --> 00:02:48,880
So that's a puzzle for you.
Yeah, this limit turns out to exist.

50
00:02:48,880 --> 00:02:50,100
And it's about 1.8.

51
00:02:50,100 --> 00:02:54,640
But what is it exactly?
What is this limit exactly equal to?

52
00:02:54,640 --> 00:02:57,930
Can you find a, a formula for this limit?

53
00:02:57,930 --> 00:03:00,740
That turns out to be an extremely
challenging question.

54
00:03:00,740 --> 00:03:02,735
But I hope it wets your appetite.

55
00:03:02,735 --> 00:03:03,281
With not

56
00:03:03,281 --> 00:03:07,259
very many tools at hand, it's possible to
build relatively

57
00:03:07,259 --> 00:03:12,910
simple seeming things that are actually
quite complicated to describe precisely.

58
00:03:12,910 --> 00:03:17,595
[SOUND]

59
00:03:17,595 --> 00:03:24,160
[MUSIC].